Isoperimetric rigidity and distributions of 1-Lipschitz functions

TL;DR

This paper establishes a new splitting theorem for metric measure spaces and weighted Riemannian manifolds based on isoperimetric conditions and the maximal observable variance of 1-Lipschitz functions, extending classical geometric results.

Contribution

It introduces a novel isoperimetric rigidity condition linked to the maximal observable variance, leading to a new splitting theorem for certain metric spaces and manifolds.

Findings

Spaces with maximal observable variance are foliated by minimal geodesics.

The results generalize Cheeger-Gromoll's splitting theorem.

A new isometric splitting theorem for weighted Riemannian manifolds is derived.

Abstract

We prove that if a geodesic metric measure space satisfies a comparison condition for isoperimetric profile and if the observable variance is maximal, then the space is foliated by minimal geodesics, where the observable variance is defined to be the supremum of the variance of 1-Lipschitz functions on the space. Our result can be considered as a variant of Cheeger-Gromoll's splitting theorem and also of Cheng's maximal diameter theorem. As an application, we obtain a new isometric splitting theorem for a complete weighted Riemannian manifold with a positive Bakry-\'Emery Ricci curvature.